Kronecker Gaussian Process - examples in Stan

These examples demonstrate Gaussian Processes (GP) regression with non-Gaussian likelihoods and missing values for Kronecker-structured covariance matrices.

Key idea

If $\mathbf{x} = (x_1, x_2)$ and the kernel factorizes as:

$$ k((x_1,x_2),(x_1',x_2')) = k_1(x_1,x_1'),k_2(x_2,x_2'), $$

then the covariance matrix over grids $X_1 \times X_2$ can be written as:

$$ K = K_2 \otimes K_1. $$

For three dimensions (e.g., space $\times$ space $\times$ time):

$$ k((x_1,x_2,t),(x_1',x_2',t')) = k_1(x_1,x_1'),k_2(x_2,x_2'),k_t(t,t'), $$

implying:

$$ K = K_t \otimes K_2 \otimes K_1. $$

This structure enables efficient computations (e.g. Cholesky) at $O(n^{3/d})$ instead of $O(n^3)$.

Likelihoods

• Normal:

$$ y(\mathbf{x}) = f(\mathbf{x}) + \epsilon, \quad \epsilon \sim \mathcal{N}(0,\sigma^2). $$

• Poisson:

$$ y(\mathbf{x}) \sim \text{Poisson}(\exp(f(\mathbf{x}))). $$

Examples

• 01: 2d GP with Normal likelihood, no missingness
• 02: 2d GP with Poisson likelihood, no missingness
• 03: 2d GP with Poisson likelihood, with missing values (impute from posterior)
• 04: 3d GP (e.g. space-time), Poisson likelihood, no missingness, fixed parameters
• 05: 3d GP with Poisson likelihood, no missingness, parameter inference
• 06: 3d GP with Poisson likelihood, with missing values (impute from posterior)